Cases where finding the minimum entropy coloring of
a characteristic graph is a polynomial time problem
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Feizi, Soheil, and Muriel Medard. “Cases Where Finding the
Minimum Entropy Coloring of a Characteristic Graph Is a Polynomial
Time Problem.” IEEE, 2010. 116–120. © Copyright 2010 IEEE
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http://dx.doi.org/10.1109/ISIT.2010.5513270
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Cases Where Finding the Minimum Entropy
Coloring of a Characteristic Graph is a Polynomial
Time Problem
Soheil Feizi, Muriel M´edardRLE at MIT
Emails:{sfeizi,medard}@mit.edu
Abstract—In this paper, we consider the problem of finding the minimum entropy coloring of a characteristic graph under some conditions which allow it to be in polynomial time. This problem arises in the functional compression problem where the computation of a function of sources is desired at the receiver. The rate region of the functional compression problem has been considered in some references under some assumptions. Recently, Feizi et al. computed this rate region for a general one-stage tree network and its extension to a general tree network. In their proposed coding scheme, one needs to compute the minimum entropy coloring (a coloring random variable which minimizes the entropy) of a characteristic graph. In general, finding this coloring is an NP-hard problem (as shown by Cardinal et al.) . However, in this paper, we show that depending on the characteristic graph’s structure, there are some interesting cases where finding the minimum entropy coloring is not NP-hard, but tractable and practical. In one of these cases, we show that, having a non-zero joint probability condition on RVs’ distributions, for any desired function f, makes characteristic graphs to be formed of some non-overlapping fully-connected maximal independent sets. Therefore, the minimum entropy coloring can be solved in polynomial time. In another case, we show that iff is a quantization function, this problem is also tractable.
Index Terms—Functional compression, graph coloring, graph entropy.
I. INTRODUCTION
While data compression considers the compression of sources at transmitters and their reconstruction at receivers, functional compression does not consider the recovery of whole sources, but the computation of a function of sources at the receiver(s).
This problem has been considered in different references (e.g., [1], [2], [3], [4]). In a recent work in [5], we showed that, for any one-stage tree network, if source nodes send -colorings of their characteristic graphs satisfying a condition called the Coloring Connectivity Condition (C.C.C.), followed by a Slepian-Wolf encoder, it will lead to an achievable coding scheme. Conversely, any achievable scheme induces colorings on high probability subgraphs of characteristic graphs satisfy-ing C.C.C. An extension to a tree network is also considered in [5].
In the proposed coding scheme in [5], we need to compute the minimum entropy coloring (a coloring random variable This work was supported by AFOSR award number FA9550-09-1-0196.
which minimizes the entropy) of a characteristic graph. In general, finding this coloring is an NP-hard problem ([6]). However, in this paper, we show that depending on the charac-teristic graph’s structure, there are certain cases where finding the minimum entropy coloring is not NP-hard, but tractable and practical. In one of these cases, we show that, having a non-zero joint probability condition on RVs’ distributions, for any desired function f, makes characteristic graphs to be formed of some non-overlapping fully-connected maximal independent sets. Then, the minimum entropy coloring can be solved in polynomial time. In another case, we show that iff is a quantization function, this problem is also tractable.
The rest of the paper is organized as follows. Section II presents the minimum entropy coloring problem statement and the necessary technical background. In Section III, main re-sults of this paper are expressed. An application of this paper’s results in the functional compression problem is explained in Section IV. We conclude the paper in Section V.
II. PROBLEMSETUP
In this section, after expressing necessary definitions, we formulate the minimum entropy coloring problem.
We start with the definition of a characteristic graph of a random variable. Consider the network shown in Figure 1 with two sources with RVs X1 and X2 such that the computation of a function (f(X1, X2)) is desired at the receiver. Suppose these sources are drawn from finite sets
X1 = {x11, x21, ..., x|X1|1 } and X2 = {x12, x22, ..., x|X2|2 }. These
sources have a joint probability distribution p(x1, xk). We express n-sequences of these RVs as X1 = {X1i}i=l+n−1i=l andX2= {X2i}i=l+n−1i=l with the joint probability distribution
p(x1, x2). Without loss of generality, we assume l = 1, and
to simplify notation, n will be implied by the context if no confusion arises. We refer to theithelement ofxjasxji. We usex1j,x2j,... as differentn-sequences of Xj. We shall drop the superscript when no confusion arises. Since the sequence (x1, x2) is drawn i.i.d. according to p(x1, x2), one can write
p(x1, x2) =ni=1p(x1i, x2i).
Definition 1. The characteristic graphGX1= (VX1, EX1) of
X1with respect to X2,p(x1, x2), and function f(X1, X2) is
defined as follows:VX1= X1and an edge(x11, x21) ∈ X12is in
EX1iff there exists ax12∈ X2such thatp(x11, x12)p(x21, x12) >
0 and f(x1
X
2f (X1,X2)
X
1Fig. 1. A network with two sources where the computation of a function is desired at the receiver.
In other words, in order to avoid confusion about the function f(X1, X2) at the receiver, if (x11, x21) ∈ EX1, then descriptions ofx11andx21must be different.
Next, we define a coloring function for a characteristic graph.
Definition 2. A vertex coloring of a graph is a function
cGX1(X1) : VX1 → N of a graph GX1 = (VX1, EX1) such
that (x11, x21) ∈ EX1 implies cGX1(x11) = cGX1(x21). The
entropy of a coloring is the entropy of the induced distribution on colors. Here, p(cGX1(xi1)) = p(cG−1X1(cGX1(xi1))), where
c−1
GX1(xi1) = {x
j
1 : cGX1(xj1) = cGX1(xi1)} for all valid j is
called a color class. We also call the set of all valid colorings of a graphGX1 asCGX1.
Sometimes one needs to consider sequences of a RV with lengthn. In order to deal with these cases, we can extend the definition of a characteristic graph for vectors of a RV. Definition 3. The n-th power of a graph GX1 is a graph
Gn
X1= (VX1n , EX1n ) such that VX1n = X1nand(x11, x21) ∈ EX1n
when there exists at least one i such that (x11i, x21i) ∈ EX1.
We denote a valid coloring ofGnX1 bycGn
X1(X1).
In some problems such as the functional compression prob-lem, we need to find a coloring random variable of a character-istic graph which minimizes the entropy. The problem is how to compute such a coloring for a given characteristic graph. In other words, this problem can be expressed as follows. Given a characteristic graphGX1 (or, its n-th power, GnX1), one can assign different colors to its vertices. SupposeCGX1 is the collection of all valid colorings of this graph, GX1. Among these colorings, one which minimizes the entropy of the coloring RV is called the minimum-entropy coloring, and we refer to it bycminGX1. In other words,
cmin
GX1 = arg minc
GX1∈CGX1H(cGX1).
(1) The problem is how to computecminGX1 givenGX1.
III. MINIMUMENTROPYCOLORING
In this section, we consider the problem of finding the minimum entropy coloring of a characteristic graph. The problem is how to compute a coloring of a characteristic graph which minimizes the entropy. In general, findingcminG
X1
is an NP-hard problem ([6]). However, in this section, we show that depending on the characteristic graph’s structure, there are some interesting cases where finding the minimum
x11 x12 x13 w1 w2 x11 x12 x13 w1 w2 (a) (b)
Fig. 2. Having non-zero joint probability distribution, a) maximal inde-pendent sets can not overlap with each other (this figure is to depict the contradiction) b) maximal independent sets should be fully connected to each other. In this figure, a solid line represents a connection, and a dashed line means no connection exists.
entropy coloring is not NP-hard, but tractable and practical. In one of these cases, we show that, by having a non-zero joint probability condition on RVs’ distributions, for any desired functionf, finding cminG
X1 can be solved in polynomial time. In
another case, we show that iff is a quantization function, this problem is also tractable. For simplicity, we consider functions with two input RVs, but one can easily extend all discussions to functions with more input RVs than two.
A. Non-Zero Joint Probability Distribution Condition
Consider the network shown in Figure 1. Source RVs have a joint probability distribution p(x1, x2), and the re-ceiver wishes to compute a deterministic function of sources (i.e., f(X1, X2)). In Section IV, we will show that in an achievable coding scheme, one needs to compute minimum entropy colorings of characteristic graphs. The question is how source nodes can compute minimum entropy colorings of their characteristic graphs GX1 and GX2 (or, similarly the minimum entropy colorings of GnX1 and GnX2, for somen). For an arbitrary graph, this problem is NP-hard ([6]). However, in certain cases, depending on the probability distribution or the desired function, the characteristic graph has some special structure which leads to a tractable scheme to find the minimum entropy coloring. In this section, we consider the effect of the probability distribution.
Theorem 4. Suppose for all(x1, x2) ∈ X1× X2,p(x1, x2) >
0. Then, maximal independent sets of the characteristic graph
GX1 (and, its n-th power GnX1, for any n) are some
non-overlapping fully-connected sets. Under this condition, the minimum entropy coloring can be achieved by assigning different colors to its different maximal independent sets.
Proof: SupposeΓ(GX1) is the set of all maximal
indepen-dent sets of GX1. Let us proceed by contradiction. Consider Figure 2-a. Suppose w1 andw2 are two different non-empty maximal independent sets. Without loss of generality, assume
x1
1andx21are inw1, andx21andx31are inw2. In other words,
these sets have a common elementx21. Sincew1 and w2 are two different maximal independent sets,x11 /∈ w2andx31 /∈ w1. Sincex11andx21are inw1, there is no edge between them in
GX1. The same argument holds forx21andx31. But, we have an
edge betweenx11andx31, becausew1andw2are two different
x21 x22 x23 x11 x12 x13 f(x1,x2) 1 1 1 2 3 3 3 4 X1 X2
Fig. 3. Having non-zero joint probability condition is necessary for Theorem 4. A dark square represents a zero probability point.
maximal independent sets, and at least there should exist such an edge between them. Now, we want to show that it is not possible.
Since there is no edge betweenx11andx21, for anyx12∈ X2,
p(x1
1, x12)p(x21, x12) > 0, and f(x11, x12) = f(x21, x12). A similar
argument can be expressed for x21 and x31. In other words, for any x12 ∈ X2, p(x21, x12)p(x31, x12) > 0, and f(x21, x12) =
f(x3
1, x12). Thus, for all x12 ∈ X2, p(x11, x12)p(x31, x12) > 0, and f(x11, x12) = f(x31, x12). However, since x11 and x31 are connected to each other, there should exist a x12 ∈ X2 such that f(x11, x12) = f(x31, x12) which is not possible. So, the contradiction assumption is not correct and these two maximal independent sets do not overlap with each other.
We showed that maximal independent sets cannot have overlaps with each other. Now, we want to show that they are also fully connected to each other. Again, let us proceed by contradiction. Consider Figure 2-b. Suppose w1 and w2
are two different non-overlapping maximal independent sets. Suppose there exists an element in w2 (call it x31) which is connected to one of elements in w1 (call it x11) and is not connected to another element of w1 (call it x21). By using a similar discussion to the one in the previous paragraph, it is not possible. Thus,x31should be connected to x21. Therefore, if for all (x1, x2) ∈ X1× X2, p(x1, x2) > 0, then maximal independent sets of GX1 are some separate fully connected sets. In other words, the complement of GX1 is formed by some non-overlapping cliques. Finding the minimum entropy coloring of this graph is trivial and can be achieved by assign-ing different colors to these non-overlappassign-ing fully-connected maximal independent sets.
This argument also holds for any power ofGX1. Suppose
x1
1, x21 and x31 are some typical sequences in X1n. If x11 is
not connected tox21andx31, it is not possible to havex21and
x3
1connected. Therefore, one can apply a similar argument to
prove the theorem for GnX1, for some n. This completes the proof.
One should notice that the conditionp(x1, x2) > 0, for all
(x1, x2) ∈ X1× X2, is a necessary condition for Theorem 4.
In order to illustrate this, consider Figure 3. In this example,
x1
1,x21andx31are inX1, andx12,x22andx32are inX2. Suppose
p(x2
1, x22) = 0. By considering the value of function f at these
points depicted in the figure, one can see that, inGX1,x21 is
not connected tox11andx31. However,x11andx31are connected to each other. Thus, Theorem 4 does not hold here.
f(x1,x2) (a) X1 1 2 3 4 5 X2 f(x1,x2) (b) 1 X2 1 4 4 2 2 5 5 3 X21 X22 X23 X11 X12 X13 X1 Function Region X12×X23
Fig. 4. a) A quantization function. Function values are depicted in the figure on each rectangle. b) By extending sides of rectangles, the plane is covered by some function regions.
It is also worthwhile to notice that the condition used in Theorem 4 only restricts the probability distribution and it does not depend on the function f. Thus, for any function
f at the receiver, if we have a non-zero joint probability
distribution of source RVs (for example, when source RVs are independent), finding the minimum-entropy coloring and, therefore, the proposed functional compression scheme of Section IV and reference [5], is easy and tractable.
B. Quantization Functions
In Section III-A, we introduced a condition on the joint probability distribution of RVs which leads to a specific structure of the characteristic graph such that finding the minimum entropy coloring is not NP-hard. In this section, we consider some special functions which lead to some graph structures so that one can easily find the minimum entropy coloring.
An interesting function is a quantization function. A nat-ural quantization function is a function which separates the
X1− X2plane into some rectangles such that each rectangle
corresponds to a different value of that function. Sides of these rectangles are parallel to the plane axes. Figure 4-a depicts such a quantization function.
Given a quantization function, one can extend different sides of each rectangle in theX1− X2plane. This may make some new rectangles. We call each of them a function region. Each function region can be determined by two subsets ofX1and
X2. For example, in Figure 4-b, one of the function regions is
distinguished by the shaded area.
Definition 5. Consider two function regions X11× X21 and X2
1 × X22. If for anyx11 ∈ X11 andx21 ∈ X12, there existx2
such thatp(x11, x2)p(x21, x2) > 0 and f(x11, x2) = f(x21, x2), we say these two function regions are pairwiseX1-proper.
Theorem 6. Consider a quantization function f such that
its function regions are pairwiseX1-proper. Then,GX1 (and Gn
X1, for any n) is formed of some non-overlapping
fully-connected maximal independent sets, and its minimum entropy coloring can be achieved by assigning different colors to different maximal independent sets.
Proof: We first prove it forGX1. SupposeX11× X21, and
X2
function f, where X11 = X12. We show that X11 and X12 are two non-overlapping fully-connected maximal independent sets. By definition,X11andX12are two non-equal partition sets ofX1. Thus, they do not have any element in common.
Now, we want to show that vertices of each of these partition sets are not connected to each other. Without loss of generality, we show it for X11. If this partition set of X1 has only one element, this is a trivial case. So, supposex11andx21are two elements in X11. By definition of function regions, one can see that, for any x2∈ X2 such thatp(x11, x2)p(x21, x2) > 0,
thenf(x11, x2) = f(x21, x2). Thus, these two vertices are not connected to each other. Now, supposex31is an element inX12. Since these function regions areX1-proper, there should exist at least onex2 ∈ X2, such that p(x11, x2)p(x31, x2) > 0, and
f(x1
1, x2) = f(x31, x2). Thus, x11andx31are connected to each
other. Therefore, X11 andX12 are two non-overlapping fully-connected maximal independent sets. One can easily apply this argument to other partition sets. Thus, the minimum entropy coloring can be achieved by assigning different colors to different maximal independent sets (partition sets). The proof forGnX1, for anyn, is similar to the one mentioned in Theorem 4. This completes the proof.
It is worthwhile to mention that without X1-proper con-dition of Theorem 6, assigning different colors to different partitions still leads to an achievable coloring scheme. How-ever, it is not necessarily the minimum entropy coloring. In other words, without this condition, maximal independent sets may overlap.
Corollary 7. If a functionf is strictly increasing (or,
decreas-ing) with respect toX1, andp(x1, x2) = 0, for all x1∈ X1 andx2 ∈ X2, then, GX1 (and, GnX1 for any n) would be a complete graph.
Under conditions of Corollary 7, functional compression does not give us any gain, because, in a complete graph, one should assign different colors to different vertices. Traditional compression in which f is the identity function is a special case of Corollary 7.
IV. APPLICATIONS IN THEFUNCTIONALCOMPRESSION
PROBLEM
As we expressed in Section II, results of this paper can be used in the functional compression problem. In this section, after giving the functional compression problem statement, we express some of previous results. Then, we explain how this paper’s results can be applied in this problem.
A. Functional Compression Problem Setup
Consider the network shown in Figure 1. In this network, we have two source nodes with input processes{Xji}∞i=1forj = 1, 2. The receiver wishes to compute a deterministic function
f : X1×X2→ Z, or f : X1n×X2n→ Zn, its vector extension.
Source node j encodes its message at a rate RXj. In other words, encoder enXj maps Xjn → {1, ..., 2nRXj}. The receiver has a decoder r, which maps {1, ..., 2nRX1} ×
{1, ..., 2nRX2} → Zn. We sometimes refer to this encod-ing/decoding scheme as ann-distributed functional code.
For any encoding/decoding scheme, the probability of error is defined as Pen = P r(x1, x2) :
f(x1, x2) = r(enX1(x1), enX2(x2))
. A rate pair,
R = (RX1, RX2) is achievable iff there exist encoders
in source nodes operating in these rates, and a decoderr at the receiver such that Pen → 0 as n → ∞. The achievable rate region is the set closure of the set of all achievable rates.
B. Prior Results in the Functional Compression Problem
In order to express some previous results, we need to define an-coloring of a characteristic graph.
Definition 8. Given a non-empty set A ⊂ X1× X2, de-fine ˆp(x1, x2) = p(x1, x2)/p(A) when (x1, x2) ∈ A, and
ˆp(x1, x2) = 0 otherwise. ˆp is the distribution over (x1, x2)
conditioned on(x1, x2) ∈ A. Denote the characteristic graph ofX1with respect toX2,ˆp(x1, x2), and f(X1, X2) as ˆGX1=
( ˆVX1, ˆEX1) and the characteristic graph of X2 with respect
toX1, ˆp(x1, x2), and f(X1, X2) as ˆGX2= ( ˆVX2, ˆEX2). Note that ˆEX1 ⊆ EX1 and ˆEX2 ⊆ EX2. Finally, we say that cGX1(X1) and cGX2(X2) are -colorings of GX1 andGX2 if
they are valid colorings of ˆGX1and ˆGX2defined with respect to some setA for which p(A) ≥ 1 − .
Among -colorings of GX1, the one which minimizes the entropy is called the minimum entropy -coloring of GX1. Sometimes, we refer to it as the minimum entropy coloring when no confusion arises.
Consider the network shown in Figure 1. The rate region of this network has been considered in different references under some assumptions. Reference [3] considered the case where
X2is available at the receiver as the side information, while
reference [4] computed a rate region under some conditions on source distributions called the zigzag condition. The case of having more than one desired function at the receiver with the side information was considered in [7].
However, the rate region for a more general case has been considered in [5]. Reference [5] shows that if source nodes send -colorings of their characteristic graphs satisfy-ing a condition called the Colorsatisfy-ing Connectivity Condition (C.C.C.), followed by a Slepian-Wolf encoder, it will lead to an achievable coding scheme. Conversely, any achievable scheme induces colorings on high probability subgraphs of characteristic graphs satisfying C.C.C.
In the following, we briefly explain C.C.C. for the case of having two source nodes. An extension to the case of having more source nodes than one is presented in [5].
Definition 9. A path with lengthm between two points Z1= (x1
1, x12) and Zm = (x21, x22) is determined by m points Zi,
1 ≤ i ≤ m such that,
i)P (Zi) > 0, for all 1 ≤ i ≤ m.
ii)ZiandZi+1only differ in one of their coordinates.
Definition 9 can be expressed for two n-length vectors (x1
1, x12) and (x21, x22).
Definition 10. A joint-coloring family JC for random vari-ables X1and X2 with characteristic graphsGX1 andGX2, and any valid coloringscGX1andcGX2, respectively is defined
x11 x12 x21 x22 f(x1,x2) 0 0 0 x11 x12 x21 x22 f(x1,x2) 1 0 (a) (b)
Fig. 5. Two examples of a joint coloring class: a) satisfying C.C.C. b) not satisfying C.C.C. Dark squares indicate points with zero probability. Function values are depicted in the picture.
as JC = {jc1, ..., jcnjc} where jci = (xi1 1, xj12), (xi21, xj22) : cGX1(xi11) = cGX1(xi21), cGX2(xj12) = cGX2(xj22) for any validi1,i2,j1andj2, wherenjc= |cGX1| × |cGX2|. We call eachjcias a joint coloring class.
Definition 10 can be expressed for RVs X1 and X2 with characteristic graphsGnX1andGnX2, and any valid-colorings
cGn
X1 andcGnX2, respectively.
Definition 11. Consider RVsX1 andX2with characteristic graphs GX1 and GX2, and any valid colorings cGX1 and cGX2. We say these colorings satisfy the Coloring Connectivity
Condition (C.C.C.) when, between any two points injci∈ JC, there exists a path that lies injci, or functionf has the same value in disconnected parts ofjci.
C.C.C. can be expressed for RVsX1 andX2with charac-teristic graphsGnX1 andGnX2, and any valid-colorings cGn
X1 andcGn
X2, respectively.
In order to illustrate this condition, we borrow an example from [5].
Example 12. For example, suppose we have two random
variables X1 and X2 with characteristic graphs GX1 and GX2. Let us assumecGX1 andcGX2 are two valid colorings
ofGX1andGX2, respectively. AssumecGX1(x11) = cGX1(x21) and cGX2(x12) = cGX2(x22). Suppose j1c represents this joint coloring class. In other words, jc1 = {(xi1, xj2)}, for all
1 ≤ i, j ≤ 2 when p(xi
1, xj2) > 0. Figure 5 considers two
different cases. The first case is whenp(x11, x22) = 0, and other points have a non-zero probability. It is illustrated in Figure 5-a. One can see that there exists a path between any two points in this joint coloring class. So, this joint coloring class satisfies C.C.C. If other joint coloring classes ofcGX1 andcGX2 satisfy C.C.C., we saycGX1 andcGX2 satisfy C.C.C. Now, consider the second case depicted in Figure 5-b. In this case, we have p(x1
1, x22) = 0, p(x21, x12) = 0, and other points have a
non-zero probability. One can see that there is no path between
(x1
1, x12) and (x21, x22) in jc1. So, though these two points belong
to a same joint coloring class, their corresponding function values can be different from each other. For this example, j1
c does not satisfy C.C.C. Therefore,cGX1 andcGX2 do not
satisfy C.C.C.
In Section III, we did not consider C.C.C. in finding the minimum entropy coloring. For the non-zero joint probability
case, under the condition mentioned in Theorem 4, any valid colorings (and therefore, minimum entropy colorings) of char-acteristic graphs satisfy C.C.C mentioned in Definition 11. It is because we have at least one path between any two points. However, for a quantization function, after achieving minimum entropy colorings, C.C.C. should be checked for colorings. In other words, despite the non-zero joint probability condition where C.C.C. always holds, for the quantization function case, one should check C.C.C. separately.
V. CONCLUSION
In this paper we considered the problem of finding the minimum entropy coloring of a characteristic graph under some conditions which make it to be done in polynomial time. This problem has been raised in the functional compression problem where the computation of a function of sources is desired at the receiver. Recently, [5] computed the rate region of the functional compression problem for a general one stage tree network and its extension to a general tree network. In the proposed coding schemes of [5] and some other references such as [4] and [7], one needs to compute the minimum entropy coloring of a characteristic graph. In general, finding this coloring is an NP-hard problem ([6]). However, in this paper, we showed that, depending on the characteristic graph’s structure, there are some interesting cases where finding the minimum entropy coloring is not NP-hard, but tractable and practical. In one of these cases, we show that, by having a non-zero joint probability condition on RVs’ distributions, for any desired functionf, finding the minimum entropy coloring can be solved in polynomial time. In another case, we show that, iff is a type of a quantization function, this problem is also tractable.
In this paper, we considered two cases such that finding the minimum entropy coloring of a characteristic graph can be done in polynomial time. Depending on different applications, one can investigate other conditions on either probability distributions or the desired function to have a special structure of the characteristic graph which may lead to polynomial time algorithms to find the minimum entropy coloring.
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